Start by combining the sweetened condensed milk and cold water in a bowl. Whisk together until smooth, then slowly whisk in the instant vanilla pudding mix. Keep whisking until the mixture is completely smooth and uniform. Cover the bowl with plastic wrap and refrigerate for at least 3 hours, allowing the pudding to set.
After the pudding has cooled and thickened, use a hand mixer or stand mixer to whip the heavy cream until it reaches a stiff peak. Carefully fold the whipped cream into the chilled pudding using a spatula, blending them until the mixture is smooth and airy.
In a bowl, sift together the flour and powdered sugar. Add the softened butter, milk, and egg yolk, and mix until the dough comes together. Knead until smooth.
Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper. Roll the dough into small balls, about 1 tablespoon each, and space them evenly on the sheet. Flatten each ball slightly. Bake for 12-15 minutes or until the cookies are golden around the edges. Let them cool completely before assembling the pudding.
In a high-sided round or rectangular dish, arrange a layer of butter cookies at the bottom, followed by a layer of sliced bananas. Spoon half of the pudding cream over the bananas and cookies, spreading it out gently with a spatula. Repeat the layers with the remaining butter cookies, bananas, and pudding cream.
To finish, garnish the top with extra banana slices and crumbled cookies, if desired. Cover and refrigerate until ready to serve. The pudding is best when it’s allowed to chill for several hours or overnight to let the flavors meld.